Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T11:02:53.307Z Has data issue: false hasContentIssue false
  • English
  • Français

AN OPTIMAL DIVIDEND POLICY WITH DELAYED CAPITAL INJECTIONS

Part of: Applications Stochastic systems and control

Published online by Cambridge University Press:  18 March 2014

ZHUO JIN  and
GEORGE YIN
ZHUO JIN*
Affiliation:
Centre for Actuarial Studies, Department of Economics, The University of Melbourne, VIC 3010, Australia
GEORGE YIN
Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, USA email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This work focuses on finding optimal dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections with delays. Starting from the classical Cramér–Lundberg process, using the dynamic programming approach, the value function obeys a quasi-variational inequality. With delays in capital injections, the company will be exposed to the risk of financial ruin during the delay period. In addition, the optimal dividend payment and capital injection strategy should balance the expected cost of the possible capital injections and the time value of the delay period. In this paper, the closed-form solution of the value function and the corresponding optimal policies are obtained. Some limiting cases are also discussed. A numerical example is presented to illustrate properties of the solution. Some economic insights are also given.

Keywords

capital injectiondelaydividend policysingular control

MSC classification

Secondary: 93E20: Optimal stochastic control 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Article
Information
The ANZIAM Journal , Volume 55 , Issue 2 , October 2013 , pp. 129 - 150
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Asmussen, S. and Taksar, M. I., “Controlled diffusion models for optimal dividend pay-out”, Insurance: Math. Econ. 20 (1997) 1–15; doi:10.1016/S0167-6687(96)00017-0.Google Scholar
Bayraktar, E. and Egami, M., “The effects of implementation delay on decision-making under uncertainty”, Stoch. Proc. Appl. 117 (2007) 333–358; doi:10.1016/j.spa.2006.08.004.CrossRefGoogle Scholar
De Finetti, B., “Su un’impostazione alternativa della teoria collettiva del rischio”, Trans. XVth Intl Congress of Actuaries 2 (1957) 433–443.Google Scholar
Dickson, D. C. M. and Waters, H. R., “Some optimal dividends problems”, ASTIN Bull. 34 (2004) 49–74; doi:10.2143/AST.34.1.504954.CrossRefGoogle Scholar
Egami, M. and Young, V. R., “Optimal reinsurance strategy under fixed cost and delay”, Stoch. Proc. Appl. 119 (2009) 1015–1034; doi:10.1016/j.spa.2008.04.005.CrossRefGoogle Scholar
Fleming, W. H. and Soner, H. M., Controlled Markov processes and viscosity solutions, 2nd edn, Volume 25 of Stochastic Modelling and Applied Probability (Springer, New York, 2006).Google Scholar
Gerber, H. U. and Shiu, E. S. W., “Optimal dividends: analysis with Brownian motion”, North Am. Actuar. J. 8 (2004) 1–20; doi:10.1080/10920277.2004.10596125.CrossRefGoogle Scholar
He, L. and Liang, Z., “Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs”, Insurance Math. Econom. 44 (2009) 88–94 ; doi:10.1016/j.insmatheco.2008.10.001.CrossRefGoogle Scholar
Jin, Z., Yang, H. and Yin, G., “Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections”, Automatica 49 (2013) 2317–2329; doi:10.1016/j.automatica.2013.04.043.CrossRefGoogle Scholar
Jin, Z., Yin, G. and Zhu, C., “Numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation”, Automatica 48 (2012) 1489–1501; doi:10.1016/j.automatica.2012.05.039.CrossRefGoogle Scholar
Kulenko, N. and Schimidli, H., “Optimal dividend strategies in a Cramér–Lundberg model with capital injections”, Insurance Math. Econom. 43 (2008) 270–278 ; doi:10.1016/j.insmatheco.2008.05.013.CrossRefGoogle Scholar
Peura, S. and Keppo, J., “Optimal bank capital with costly recapitalization”, J. Business 79 (2006) 2163–2201; doi:10.1086/503660.CrossRefGoogle Scholar
Reddemann, S., Basse, T. and Graf von der Schulenburg, J.-M., “On the impact of the financial crisis on the dividend policy of the European insurance industry”, Geneva Papers on Risk and Insurance: Issues and Practice 35 (2010) 53–62; doi:10.1057/gpp.2009.37.CrossRefGoogle Scholar
Sethi, S. P. and Taksar, M. I., “Optimal financing of a corporation subject to random returns”, Math. Finance 12 (2002) 155–172; doi:10.1111/1467-9965.t01-2-02002.CrossRefGoogle Scholar
Thonhauser, S. and Albrecher, H., “Optimal dividend strategies for a compound Poisson process under transaction costs and power utility”, Stoch. Models 27 (2011) 120–140 ; doi:10.1080/15326349.2011.542734.CrossRefGoogle Scholar
Yao, D., Yang, H. and Wang, R., “Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs”, Euro. J. Operat. Res.(211) (2011) 568–576 ; doi:10.1016/j.ejor.2011.01.015.CrossRefGoogle Scholar
Yin, G., Jin, Z. and Yang, H., “Asymptotically optimal dividend policy for regime-switching compound Poisson models”, Acta Math. Appl. Sin. Engl. Ser. 26 (2010) 529–542 ; doi:10.1007/s10255-010-0023-0.CrossRefGoogle Scholar